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Theorem ndmovcom 6007
Description: Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
Hypothesis
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
Assertion
Ref Expression
ndmovcom  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  ( B F A ) )

Proof of Theorem ndmovcom
StepHypRef Expression
1 ndmov.1 . . 3  |-  dom  F  =  ( S  X.  S )
21ndmov 6004 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
3 ancom 437 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  ( B  e.  S  /\  A  e.  S )
)
41ndmov 6004 . . 3  |-  ( -.  ( B  e.  S  /\  A  e.  S
)  ->  ( B F A )  =  (/) )
53, 4sylnbi 297 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( B F A )  =  (/) )
62, 5eqtr4d 2318 1  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   (/)c0 3455    X. cxp 4687   dom cdm 4689  (class class class)co 5858
This theorem is referenced by:  addcompi  8518  mulcompi  8520  addcompq  8574  addcomnq  8575  mulcompq  8576  mulcomnq  8577  addcompr  8645  mulcompr  8647  addcomsr  8709  mulcomsr  8711  addcomgi  27661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861
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